Abstract

We study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon equation.

1. Introduction

It is well known that finding exact travelling wave solutions of nonlinear partial differential equations (NLPDEs) is useful in many scientific applications such as fluid mechanics, plasma physics, and quantum field theory. Due to these applications many researchers are investigating exact solutions of NLPDEs since they play a vital role in the study of nonlinear physical phenomena. Finding exact solutions of such NLPDEs provides us with a better understanding of the physical phenomena that these NLPDEs describe. Several techniques have been presented in the literature to find exact solutions of the NLPDEs. These include the homogeneous balance method, the Weierstrass elliptic function expansion method, the -expansion method, the -expansion method, the Exp-function method, the tanh function method, the extended tanh function method, and the Lie group method [1–10].

In this work, we study one such NLPDE, namely, the generalized double sinh-Gordon equation:
which appears in many scientific applications [11–13]. It should be noted that when , , and , (1) becomes the generalized sinh-Gordon equation [14, 15]. Furthermore, if and , (1) reduces to the sinh-Gordon equation [16].

Many authors have studied the generalized double sinh-Gordon equation (1). Travelling waves solutions of (1) were obtained in [11] by using the tanh function method and the variable separable method. In [12] the method of bifurcation theory of dynamical system was used to prove the existence of periodic wave, solitary wave, kink and antikink wave, and unbounded wave solutions of (1). It should be noted that solutions obtained in [12] were different the ones obtained in [11]. Recently, solitary and periodic waves solutions of (1) were found in [13] by employing -expansion method. It is further shown in [13] that solutions obtained by using the -expansion method are more general than those given in [11], which were obtained by tanh function method.

In this paper, we employ an entirely different method, known as the Exp-function method, to obtain new exact solutions of the generalized sinh-Gordon equation (1). The paper is structured as follows. In Section 2, we obtain exact solutions of the generalized double sinh-Gordon equation (1) with the help of the Exp-function method. In Section 3 we present concluding remarks.

In this section we employ the Exp-function method to solve the generalized double sinh-Gordon equation (1). This method was introduced by He and Wu [17]. The Exp-function method results in the travelling wave solution based on the assumption that the solution can be expressed in the following form:
where , , , and are positive integers that can be determined and and are unknown constants. According to Exp-function method, we introduce the travelling wave substitution , where . Then (1) transforms to the nonlinear ordinary differential equation:
Further, using the transformation on (3), we obtain
We assume that the solution of (4) can be expressed as
The values of and , and can be determined by balancing the linear term of the highest order with the highest order of nonlinear term in (4), that is, and . By straight forward calculation, we have
where are coefficients only for simplicity. Balancing the highest order of Exp-function in (6), we have , which yields . Similarly, we balance the lowest order in (4) to determine values of and . We have
where are coefficients only for simplicity. Balancing the lowest order of Exp-function in (7), we have , which yields . For simplicity, we first set and . then (5) reduces to

Inserting (8) into (4) and using Maple, we obtain
where
Equating the coefficients of in (9) to zero, we obtain a set of algebraic equations:
Solving the system (11) with the help of Maple, we obtain the following three cases.

Case 1. We have the following:

Case 2. We have the following:

Case 3. We have the following:
where .

Substituting values from (12) into (8), we obtain
As a result one of the solutions of (1) is given by
where , and .

As a special case, if we choose and in (16), then we get , and obtain the solution of the generalized sinh-Gordon equation as
which is the solution obtained in [14, 15].

Now substituting the values from (13) (Case 2) into (8) results in the second solution of (1) as
with , , and .

The third solution of (1) is obtained by using the values from (14) (Case 3) and substituting them into (8). Consequently, it is given by
where , , , and .

To construct more solutions of (1), we now set and . Then (5) reduces to
Proceeding as above, we obtain the following three solutions of (1):
where , ,
with , , and , and
where , , and .

By taking , , , , and in the solution (16), we have its profile given in Figure 1.

3. Concluding Remarks

In this paper we obtained new exact solutions of the generalized double sinh-Gordon equation (1) using the Exp-function method. We presented six different solutions of (1). Earlier, the tanh function, the bifurcation, and the -expansion methods [11–13] were employed to obtain exact solutions of (1). The solutions obtained in this paper were new and were different from the ones obtained in [11–13]. By taking special values of the constants, we also retrieved the solution of the generalized sinh-Gordon equation, which was obtained in [14, 15]. The Exp-function method is very simple and straightforward method for solving nonlinear partial differential equations. Indeed this has some pronounced merit as compared to the other methods. The correctness of the solutions obtained here has been verified by substituting them back into (1).

Acknowledgments

Gabriel Magalakwe would like to thank SANHARP, NRF, and North-West University, Mafikeng Campus, South Africa, for their financial support.